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train/img_00000400.png | \begin{align*}\begin{aligned}\mathbf{y}_{k} &= \sum_{u=1}^{N} a_{u}[\mathbf{x}_{u}^{c}[k],\mathbf{x}_{u}^{c}[k+ K\cdot 1],\cdots, \mathbf{x}_{u}^{c}[k+K(L-1)] ]^{T} + \mathbf{n}_{k} \\&= \mathbf{S}\mathbf{X}_{k}\mathbf{a} + \mathbf{n}_{k},\end{aligned}\end{align*} |
train/img_00000401.png | \begin{align*}m=\max_{\overline{\Omega _{d}}}\xi \left( x\right) \Rightarrow \max_{\overline{\Omega _{d}}}\varphi _{\lambda }\left( x\right) =e^{\lambda m}.\end{align*} |
train/img_00000402.png | \begin{align*} \Delta ^{I}n_{3}=\Delta ^{I}\Big(-\frac{1}{\sqrt{g}}\Big)=\lambda _{31}\Big(-\frac{au}{\sqrt{g}}\Big)+\lambda _{32}\Big(-\frac{bv}{\sqrt{g}}\Big)+\lambda _{33}\Big(\frac{1}{\sqrt{g}}\Big).\end{align*} |
train/img_00000403.png | \begin{align*} \Gamma_+:=\omega\times \{1\}, \Gamma_-:= \omega\times \{-1\},\Gamma_0:=\gamma_0\times[-1,1]. \end{align*} |
train/img_00000404.png | R _ { 2 } ^ { 1 } = - 2 i \frac { d g _ { 1 } \wedge d \overline { { { g } } } _ { 1 } } { ( 1 - g _ { 1 } \overline { { { g } } } _ { 1 } ) ^ { 2 } } , R _ { 4 } ^ { 3 } = + 2 i \frac { d g _ { 2 } \wedge d \overline { { { g } } } _ { 2 } } { ( 1 + g _ { 2 } \overline { { { g } } } _ { 2 } ) ^ { 2 } } , |
train/img_00000405.png | E [ \sigma ] = \int _ { 0 } ^ { \infty } \epsilon ( r ) d r \, \, . |
train/img_00000406.png | \begin{align*} M_{(D)}^2 = \frac{2 M_{(D+1)}^3}{(D-2)k} \left( 1 - e^{-\frac{(D-2)}{2}\phi} \right),\end{align*} |
train/img_00000407.png | \begin{align*} \sum_{k=1}^K\sum_{j=1}^{J_k} \tilde x^{*,1}_{j,k}\bigl(\bigl(\nu^*\bigr)^-\bigr) \geq \min (\alpha, \overline\alpha) \quad\mbox{and}\sum_{k=1}^K\sum_{j=1}^{J_k} \tilde x^{*,1}_{j,k}\bigl(\bigl(\nu^*\bigr)^+\bigr) \leq\min(\alpha, \overline\alpha).\end{align*} |
train/img_00000408.png | \begin{align*}\epsilon(x^i)\equiv\epsilon(i)\,,\quad\epsilon(\xi_{1i})=\epsilon(i)+1\,.\end{align*} |
train/img_00000409.png | d s ^ { 2 } = d t ^ { 2 } - a ^ { 2 } ( t ) d \Sigma ^ { 2 } = a ^ { 2 } ( \eta ) [ d \eta ^ { 2 } - d \Sigma ^ { 2 } ] |
train/img_00000410.png | \begin{align*} -\lambda _{12}\sqrt{1+bv^{2}}+\lambda _{13}v = 0.\end{align*} |
train/img_00000411.png | \begin{align*} G(y) := \sum_{k\in [m]} \frac{\alpha_k-\delta_{1,k}}{k+1} y^{k+1} + \log(D).\end{align*} |
train/img_00000412.png | \begin{align*} \partial_r u_r + \frac{1}{r} u_r + \partial_z u_z \,=\, 0~, \partial_z u_r - \partial_r u_z \,=\, \omega_\theta~,\end{align*} |
train/img_00000413.png | \begin{align*}\zeta_k(t) = {\tilde{\zeta}}_k(t) \mbox{ for all } t \in [\tau,\tau+\delta], k \in \mathbb{N}.\end{align*} |
train/img_00000414.png | \begin{align*}\varphi\left(e_{i}\right) & =f_{i}, & \varphi(f_{i}) & =e_{i}, & \varphi(t_i) & =t_i\end{align*} |
train/img_00000415.png | E _ { \mathrm { v d W } } = { \frac { 2 3 } { 1 5 3 6 \pi a } } ( \epsilon - 1 ) ^ { 2 } . |
train/img_00000416.png | \begin{align*}2c=2av^2+\delta-\delta^q-\gamma(2av+b).\end{align*} |
train/img_00000417.png | \begin{align*}\mathbb{E}\left[\left( H - \int M(ds,\theta^H_s)\right) \int \frac{\partial }{\partial x}M(ds,\theta^H_s)\right]=0.\end{align*} |
train/img_00000418.png | \begin{align*}P_{N,m}=N_{m}+N_{-m}\ \textrm{and}\ Q_{N,m}=N_{m}-N_{-m}.\end{align*} |
train/img_00000419.png | ( 2 \partial _ { + } \partial _ { - } - \Delta _ { \perp } + m ^ { 2 } ) { \phi } = 0 . |
train/img_00000420.png | \begin{align*}f(M_n)=f(M)\cap M^\prime_n\;\;\forall n\in \mathbb{Z}^+\end{align*} |
train/img_00000421.png | \begin{align*}\alpha_{g}(\pi(x))=\begin{cases}1 \\\delta_{g}=\delta_{f} \end{cases}\end{align*} |
train/img_00000422.png | \begin{gather*}P({\cal M}{\rm YB}_n,s,t)=\sum_{\pi} s^{\#(\pi)} t^{n(\pi)},\end{gather*} |
train/img_00000423.png | \begin{align*}a_{k}^{\dagger}T_{k}=q_k^{-1}T_{k}a_k^{\dagger},\end{align*} |
train/img_00000424.png | \begin{align*}z \;=\; e^{\frac{\pi}{N}w} \;=\; e^{\frac{\pi}{N}(t+i\theta)}\end{align*} |
train/img_00000425.png | \begin{align*}\tilde{f}(s,x,t,y)=f(-t,y,-s,x)\end{align*} |
train/img_00000426.png | \begin{align*}\nu([g,f])& \leq \nu(g[f,hh_0h^{-1}]g^{-1})+\nu([f,hh_0h^{-1}]^{-1})\\&=2\nu([f,hh_0h^{-1}])\\ & \leq 2(\nu(f(hh_0h^{-1})f^{-1})+\nu((hh_0h^{-1})^{-1}))\\ &=4\nu(hh_0h^{-1})=4\nu(h_0).\end{align*} |
train/img_00000427.png | \begin{align*}\rho^2(3s.sin\theta -3tcos\theta+lsin\theta cos(2\theta-\omega)-hcos\theta sin(2\theta-\varphi))+ \\2\rho(kcos\theta sin(\theta+\mu)-msin\theta cos(\theta-\alpha))-2fcos\theta +2dsin\theta=0 \end{align*} |
train/img_00000428.png | \left( \frac { d ^ { 2 } } { d \phi ^ { 2 } } + \frac { 2 \nu + 1 } { \phi } \frac { d } { d \phi } - \frac { n ^ { 2 } } { g ^ { 4 } } \right) f _ { \nu } = 0 ~ . |
train/img_00000429.png | \begin{align*}\varepsilon:\widetilde{\mathfrak{osp}}_{m|2n}(R,{}^-)\rightarrow R_-,\begin{pmatrix}A&B&-\overline{\rho(C)}^t\\C&D_{11}&D_{12}\\\overline{\rho(B)}^t&D_{21}&-\overline{D}_{11}^t\end{pmatrix}\mapsto\mathrm{Tr}(A)-\mathrm{Tr}(\rho(D_{11})-\overline{\rho(D_{11})}).\end{align*} |
train/img_00000430.png | \begin{align*}\sum_{(j_1, \ldots, j_n) \in i(D'_\pi)} \varphi_q\left(z_1\left(j_{s^{-1}_\chi(1)}, j_{s^{-1}_\chi(1) + 1}; k_1\right) \cdots z_n\left(j_{s^{-1}_\chi(n)}, j_{s^{-1}_\chi(n) + 1}; k_n\right) \right) = 0.\end{align*} |
train/img_00000431.png | \begin{align*}\sum_{i=0}^{n} k(a_{n+1},a_{i})=\inf_{x\in A} \sum_{i=0}^{n} k(x,a_{i})\end{align*} |
train/img_00000432.png | \begin{align*} x_0 = \begin{pmatrix} 2\pi \sqrt{-1} & 0 \\ 0 & -2\pi \sqrt{-1} \end{pmatrix}\end{align*} |
train/img_00000433.png | \begin{align*}|T_3|\ge |A_{123}\cup A_3|+|S_3|-N/2=|A_{123}|+2|A_3|-|U''_3|+|S'_{3}|-N/2\;,\end{align*} |
train/img_00000434.png | a ^ { \prime } = W a W ^ { \dagger } , \qquad a ^ { \prime \dagger } = W a ^ { \dagger } W ^ { \dagger } . |
train/img_00000435.png | \begin{align*}x_{\mathbf{N}}^n+a_{n-1}x_{\mathbf{N}}^{n-1}+\dots +a_{k+1}x_{\mathbf{N}}^{k+1}=a_k x_{\mathbf{N}}^k+a_{k-1} x_{\mathbf{N}}^{k-1}+\dots+ a_l x_{\mathbf{N}}^l\end{align*} |
train/img_00000436.png | f _ { \nu } ^ { \prime } ( z ) = \log ( 4 a \nu ^ { 4 } ) - [ \psi ( 2 \nu - z ) + \psi ( - z ) + \psi ( 2 + 2 \nu + z ) + \psi ( 2 + 4 \nu + z ) ] , |
train/img_00000437.png | \begin{align*} \frac{d}{dz}\left[{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z) \right] &= \frac{ba}{c} \left[\sum_{n=0}^{\infty} (a+1)_n \frac{{}^{M\!L\!\!}B_{p}(b+n+1,\;c-b)}{B(b+1,\;c-b)} \frac{z^n}{n!} \right]& \\ &= \frac{ba}{c}{}^{M\!L\!}\textrm{F}_{p}(a+1,b+1;c+1;z).&\end{align*} |
train/img_00000438.png | \begin{align*}\theta = \log| \omega(X_1,\ldots,X_k,X_0)| = \log |\omega(X_1,\ldots,X_k,X_0')| = \theta'.\end{align*} |
train/img_00000439.png | \begin{align*}\langle n \| S_j({\mathsf s})\| n'\rangle=S_j({\mathsf s})= \left\{ \begin{array}{lll} &2ia_j({\mathsf s})+1 \quad &\mbox{for elastic scattering $n=n '$} \\ &2ia_j^{(n)}({\mathsf s}) \quad &\mbox{for reaction from $n'$ into the channel $n$} \, , \end{array} \right. \end{align*} |
train/img_00000440.png | \begin{align*} \langle v_t,\phi^*\rangle_m + \int_s^t \langle \psi_0(\cdot ,v_t) , \phi^*\rangle_m dr = \langle v_s,\phi^*\rangle_m \in [0,\infty), s, t > 0. \end{align*} |
train/img_00000441.png | \begin{align*}B_{j}u=\sum\limits_{\left\vert \beta \right\vert \leq l_{j}}\ b_{j\beta}\left( y\right) D_{y}^{\beta }u\left( x,y\right) =0x\in R^{n} \end{align*} |
train/img_00000442.png | \begin{align*}B_{n}(x_1,x_2,\ldots,x_n) = \sum_{\pi(n)} \frac{n!}{\prod_{r=1}^n k_r!}\prod_{r=1}^{n} \left(\frac{x_r}{r!}\right)^{k_r},\end{align*} |
train/img_00000443.png | \begin{align*}w=Uv^*,z=\|v^*\mathbf n\|^{\dagger 2}\langle M_0v^*\mathbf n,{}\cdot{}\rangle \mathbf n -G_{0,0}v,\end{align*} |
train/img_00000444.png | \begin{align*}P={\bigoplus\limits_{i=1}^{d}}\mathbb{Z\omega}_{i}.\end{align*} |
train/img_00000445.png | \begin{align*}{n_p^\ast} =\left\{\begin{aligned}&n_p^{\mathrm{ceil}}, &~\gamma_{}^{\mathrm{ceil}}\geq \gamma_{}^{\mathrm{floor}},\\&n_p^{\mathrm{floor}},&~\gamma_{}^{\mathrm{ceil}}< \gamma_{}^{\mathrm{floor}},\end{aligned}\right.\end{align*} |
train/img_00000446.png | \begin{align*}Z = \int D\varphi^i J(\varphi)\exp\left\{i\int dx\,\frac{1}{2}\partial_\mu\varphi^ig_{ij}\partial_\mu\varphi^j\right\}\equiv\int D\varphi^i J(\varphi) \exp\left\{iS_{\rm ch}(\varphi)\right\}.\end{align*} |
train/img_00000447.png | \begin{align*}<\psi |\psi >=\int d^{d-1}\mathbf{r\,\,}\psi ^{*}\left( t,\mathbf{r}\right)\psi \left( t,\mathbf{r}\right)\end{align*} |
train/img_00000448.png | \begin{align*}C_+\gamma^{(j)}C_+=(-1)^{n+1}\overline{\gamma^{(j)}},\mbox{for \ }j=1,2,\ldots,2n,\end{align*} |
train/img_00000449.png | \begin{align*}\int_0^{\infty}e^{-\frac{a^2t^2}{2}\, -\frac{b^2}{2t^2}}dt=\sqrt{\frac{\pi}{2}}\frac{1}{a}e^{-ab}.\end{align*} |
train/img_00000450.png | \begin{align*}\delta R_{MN} (h) + {1 \over 6} \bar F_{\theta\phi} \bar F^{\theta\phi} (h_{MN} + 2 \bar g_{MN} {\cal F})- ( \bar F_{M \theta} \bar F_N^{~\theta} + \bar F_{M \phi}\bar F_N^{~\phi} ) {\cal F}= 0,\end{align*} |
train/img_00000451.png | \begin{align*}P^*(\Phi) = \lim_{N \to +\infty}P^*_N(\Phi) = \sup_{N > 0}P^*_N(\Phi).\end{align*} |
train/img_00000452.png | \begin{align*}D \triangleq \left\{t \geq 0 \colon P\big(X_{t \wedge \tau_{\lambda_{m}}} \not = X_{(t \wedge \tau_{\lambda_{m}})-}\big) = 0\right\}.\end{align*} |
train/img_00000453.png | \begin{align*} A(t) &= \frac{1}{2}((A(t)+B(t))-(B(t)-A(t)))\\ &= \varphi(t^4)^2(2tH(t)H(t^4))\\ &= 2t\varphi(t^4)^2H(t^4)H(t)\end{align*} |
train/img_00000454.png | \begin{align*}U_j(X,t) = \frac{U(r_jX, r_j^2 t)}{ \bigg(\frac{1}{r_j^{n+3+a}}\int_{\mathbb Q_{r_j}^+} U^2 x_{n+1}^a dX dt\bigg)^{1/2}},\,\,\,\ \ \ \ j\geq 1. \end{align*} |
train/img_00000455.png | \begin{align*}\begin{array}{ccc}\chi ^a(\sigma )=\frac 1{\sqrt{2\pi }}\sum\limits_{r\in Z_0+\phi }\left(c_r\cos {r\sigma }+d_r\sin {r\sigma }\right) & & \\ \overline{\chi }^a(\sigma )=\frac 1{\sqrt{2\pi }}\sum\limits_{r\in Z_0+\phi}\left( \overline{c}_r\cos {r\sigma }+\overline{d}_r\sin {r\sigma }\right) & & \end{array... |
train/img_00000456.png | \begin{align*}H(t) = \int d^3x \left\{-\frac{\hbar^2}{2 a^3(t)}\frac{\delta^2}{\delta\eta^2}+\frac{a(t)}{2}\left(\vec{\nabla}\eta\right)^2+a^3(t)\left(V(\phi)+V'(\phi)\eta+\frac{1}{2}V''(\phi)\eta^2+\cdots \right)\right\} \end{align*} |
train/img_00000457.png | W ^ { ( 0 ) } ( \hat { t } , \hat { g } ) = \min _ { \{ h , \psi \} } \left( \Gamma ^ { ( 0 ) } ( h , \psi ) - \frac { 1 } { 1 2 \pi } \int h \hat { t } - \frac { 1 } { 3 \pi } \int \hat { g } \psi \right) \, . |
train/img_00000458.png | \begin{align*} \lim_{m\to+\infty}\Big(1- \Theta_m(c,c')\Big)=1\end{align*} |
train/img_00000459.png | \begin{align*}\dot{x} = X_{H_t} (x),\end{align*} |
train/img_00000460.png | \begin{align*}k+p_{n+1}=k+p_n+M(m_{n+1})\geq k+p_n+m_{n+1}\geq k_n+ m_{n+1}=k_{n+1},\end{align*} |
train/img_00000461.png | \begin{align*}J_3(\lambda) = \left[\begin{array}{ccc} \lambda & 1 & 0\\ 0 & \lambda &1 \\ 0 & 0 & \lambda\end{array}\right].\end{align*} |
train/img_00000462.png | \begin{align*} C_{d,0,n} &= \{ (f, c_1, \ldots, c_1, c_2, \ldots, c_2) \in B^{crit}_{d,k} \;\vert\; c_1 f^n(c_1) = c_1\} \\ C_{d,1,m} &= \{ (f, c_1, \ldots, c_1, c_2, \ldots, c_2) \in B^{crit}_{d,k} \;\vert\; c_2 f^m(c_2) = c_2\}. \end{align*} |
train/img_00000463.png | \begin{align*}\sum_{n=-\infty}^{\infty} e^{-{4\pi^{2}(n+1/2)^{2} \over L^{2}}t}={L \over \sqrt{4\pi t}}\Bigl( 1+2\sum_{n=1}^{\infty} (-1)^{n}e^{-{L^{2}n^{2} \over 4t}}\Bigr),\end{align*} |
train/img_00000464.png | \begin{align*} 0\geqslant \Re \langle P_n x,H_nP_n x\rangle = \Re \langle P_n x, P_n Hx\rangle = \Re \langle P_n x, Hx\rangle \to \Re \langle x,Hx\rangle \end{align*} |
train/img_00000465.png | \begin{align*}&\int_{ \{z\in M:-\Psi(z)\in V\}}|F_{t_0,\tilde{c}}|^2_h\\= & \int_{\{z\in M:-\Psi(z)\in V\}}|\tilde F|^2_h+\int_{\{z\in M:-\Psi(z)\in V\backslash N\}}|F_{t_0,\tilde{c}}-\tilde F|^2_h\\&+\int_{ \{z\in M: -\Psi(z)\in V\cap N\}}|F_{t_0,\tilde{c}}|^2_h\end{align*} |
train/img_00000466.png | \begin{align*}\begin{array}{c}Z_{N}(+\infty )=2\pi I_{max}+\pi +\zeta _{+}\\Z_{N}(-\infty )=2\pi I_{min}-\pi -\zeta _{-}\end{array}\end{align*} |
train/img_00000467.png | \mathrm { t g } \left( \frac { z _ { 0 } } { 2 { \hbar } c } \right) = - \frac { i } { 2 } . |
train/img_00000468.png | \begin{align*} \frac{i}{2\pi}\int_{\mathcal{C}}e^{-t\tau}\operatorname{Tr}\phi_{-1}\,d\tau = e^{-t\alpha |\xi|}+(n-2)e^{-t\mu|\xi|}+e^{-2t\mu|\xi|}+e^{-2t\mu s_1|\xi|}.\end{align*} |
train/img_00000469.png | \begin{align*}H_0 = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \end{pmatrix} = H_0^*, V = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix} = V^*. \end{align*} |
train/img_00000470.png | \begin{align*}\begin{aligned}\mathsf{T}_{1}(t_{k})&=\frac{2(\lambda-\sum_{i=1}^{n}\beta_{i}u_{i}(t_{k}))}{\sum_{i=1}^{n}\alpha_{i}}, \\u_i(t_{k+1})&=\beta_{i}u_i(t_{k})+\alpha_{i}\mathsf{T}_{1}(t_{k}).\end{aligned}\end{align*} |
train/img_00000471.png | \begin{align*}\left\vert 1-\left\vert \frac{k_{21}k_{13}}{k_{23}k_{11}}\right\vert\right\vert =\left\vert 1-\left\vert \frac{\widehat{k_{21}}\widehat{k_{13}}}{\widehat{k_{23}}}\right\vert \right\vert \leq\delta_{12}\delta_{13},\end{align*} |
train/img_00000472.png | \begin{align*} \frac{e^\eta-1}{\eta}\cdot e^{|\xi-\eta|} = \frac{e^\eta-1}{\eta}\cdot e^{\xi-\eta} = \frac{e^\xi-e^{|\xi-\eta|}}{\eta} \geq \frac{e^\xi-1}{\xi} \, . \end{align*} |
train/img_00000473.png | x = \left( \begin{array} { c c } { { x _ { 1 1 } } } & { { x _ { 1 2 } } } \\ { { x _ { 2 1 } } } & { { x _ { 2 2 } } } \\ \end{array} \right) , |
train/img_00000474.png | \begin{align*}*^{-1}\frac{\partial}{\partial \xi_i}*=\eta^{i\bar{j}}\bar{\xi}_j, && *^{-1}\frac{\partial}{\partial \bar{\xi}_i}*=\eta^{\bar{i}j}\xi_j, && *^{-1}\xi_i* =\eta_{\bar{j}i} \frac{\partial}{\partial \bar{\xi}_j}, && *^{-1}\bar{\xi}_i* =\eta_{j\bar{i}} \frac{\partial}{\partial \xi_j},\end{align*} |
train/img_00000475.png | \begin{align*} \left( \begin{matrix} \frac{\partial }{\partial x} & -\frac{\partial }{\partial y} \\ \frac{\partial }{\partial y} & \frac{\partial }{\partial x} \\\end{matrix} \right)\left( \begin{matrix} u \\ v \\\end{matrix} \right)=0\Leftrightarrow \frac{\partial w}{\partial \overline{z}}=0\end{align*} |
train/img_00000476.png | \begin{align*} H^0\left(G_\Sigma,D^*_{\rho_{\pmb{1},2}}\right)^\vee = H^0\left(G_\Sigma, \hat{R[[\Gamma]]}(\tau^{-1} \varkappa \kappa) \right)^\vee \cong \frac{R[[\Gamma]]}{\left(\tau^{-1} \varkappa \kappa(\gamma_0)-1\right)}.\end{align*} |
train/img_00000477.png | \begin{align*}v^2=\alpha\left[\frac{\alpha\beta \lambda }{2\pi}-2\right]c^2\end{align*} |
train/img_00000478.png | \begin{align*}S_{1}=\left\{\sum_{i=1}^{n} a_{i} \alpha_{i} \mid 0 \leq a_{i}<p q, \ a_{i} \in \mathbb{Z},\ 1 \leq i \leq n \right\}.\end{align*} |
train/img_00000479.png | \begin{align*}f_{\alpha,\beta,\lambda}(x)=|x|^{-\frac{N-1}{2}}h^{-\frac{1}{4}}(|x|)\exp \left\{-\int_R^{|x|}h^{\frac{1}{2}}(s)ds -\int_R^{|x|}v_\lambda(s)ds \right\}\;,\end{align*} |
train/img_00000480.png | \langle T _ { x ^ { - } , x ^ { - } } \rangle = - \, \frac { N } { 1 2 \pi } \left( \, \partial _ { - } ^ { 2 } \rho - ( \partial _ { - } \rho ) ^ { 2 } + t _ { - } ( x ^ { - } ) \, \right) \, . |
train/img_00000481.png | \begin{align*} M(x,e_\alpha) = \int_{-h/2}^{h/2} x_3 e_3 \times t (x,x_3,e_\alpha) dx_3, \alpha=1,2,\end{align*} |
train/img_00000482.png | \begin{align*}X_t=X_0+ \int_0^t\sigma(X_s)dB_s + \int_0^t b(X_s)ds, t < \zeta,\end{align*} |
train/img_00000483.png | \begin{align*}{\cal N}(\xi^C,\xi^A) = (\xi^C)^3 - {3 \over 2} \xi^C (\xi^A)^2 \ ,\end{align*} |
train/img_00000484.png | V _ { 1 } ( \mathrm { x ) = \frac { 1 } { 3 2 } \mathrm { x ^ { 6 } - \frac { \ o m e g a } { 1 6 \mathrm { g } } \mathrm { x ^ { 4 } + ( \frac { \ o m e g a ^ { 2 } } { 3 2 \mathrm { g ^ { 2 } } } - \frac { 3 j } { 4 } - \frac { 3 } { 8 } ) \mathrm { x ^ { 2 } - \frac { \ o m e g a } { 4 \mathrm { g } } ( 3 j - \frac ... |
train/img_00000485.png | \begin{align*}\sum_{1\le n\le t} \widetilde{G}_{n,n}(t)&=\sum_{1\le n\le t} \frac{1}{\sqrt{n+1}} \int_n^{n+1} v^{-1/2} \exp\left(-t\left(i\log{\frac{n+1}{v}}+\frac{1}{2}\log^2{\frac{n+1}{v}}\right)\right) dv \\&=\sum_{1\le n\le t} \int_1^{1+1/n} v^{-3/2} \exp\left(-t\left(i\log v+\frac{1}{2}\log^2v\right)\right) dv.\en... |
train/img_00000486.png | \begin{align*}\| Q_L u \|_{L_t^p L_{x,y}^q} \lesssim L^{\frac{2}{3 p} + \frac{1}{q}} \|Q_L u\|_{L_{x,y,t}^2}, \textnormal{if} \ \ \frac{2}{p} + \frac{2}{q} = 1, \ p \geq 4.\end{align*} |
train/img_00000487.png | \begin{align*} u(y) - u(x) &\leq h(y) = \sup_{l\in \partial_{L_x} f(x)} l(y) \\ &\leq \sup_{l\in \partial_{L_x} f(x)} f(y) - f(x) + l(x) \\ &= f(y) - f(x) \end{align*} |
train/img_00000488.png | \begin{align*}\begin{aligned} & (1+\lambda)\|\partial_t U\|_{ L_p (Q_r) } \le (1+\lambda) \big(\|\partial_t U - v \cdot D_x U\|_{ L_p (Q_r) } + r \|D_x U\|_{ L_p (Q_r) }\big) \\ & \le N (1+\lambda) (\|U\|_{ L_p (Q_{r_1}) } + \|D^{|\alpha| - 1}_v D^{1 +|\beta|}_x u\|_{L_p(Q_{r_1})} \\ &+ \|D_x U\|_{ L_p (Q_{r})}) \le N ... |
train/img_00000489.png | \begin{align*}{\displaystyle \mathbf{A}(\cdot,-\Delta t)=\mathbf{A}(\cdot,0)-\Delta t \frac{\partial \mathbf{A}}{\partial t}(\cdot,0)=\mathbf{A}_{0}-\Delta t\mathbf{A}_{1}},\end{align*} |
train/img_00000490.png | \begin{align*}S\psi =\psi ,\,\,S=J\Delta ^{\frac{1}{2}},\,J=S_{scat}J_{0} \end{align*} |
train/img_00000491.png | \begin{align*} (-(b_1^2 - d_1^2 J_1) J_1, J_2)_F = (d_1^2 J_1^2 - b_1^2 J_1, J_1)_F = 1.\end{align*} |
train/img_00000492.png | \begin{align*}L & =\frac{1}{4}F_{\sigma\tau}R^{\tau\sigma}-\frac{4\pi}{c}j^{\sigma}A_{\sigma}\\& =\frac{1}{4}F_{\sigma\tau}\epsilon^{\tau\sigma\lambda\rho}F_{\lambda\rho}-\frac{4\pi}{c}j^{\sigma}A_{\sigma},\end{align*} |
train/img_00000493.png | S ( B _ { f } ) = \frac { 1 } { 8 K } \psi \left\{ \widetilde { R } ^ { + } g \, R , \gamma ^ { 0 } \right\} = i n v , |
train/img_00000494.png | \left[ p ^ { \mu } \partial _ { \mu } + \vec { Q } \cdot \vec { F } _ { \mu \nu } p ^ { \mu } \partial _ { p } ^ { \nu } + p ^ { \mu } \vec { Q } \times \vec { A } _ { \mu } \cdot { \frac { \partial } { \partial \vec { Q } } } \right] f ( x , p , \vec { Q } ) = C ( x , p , \vec { Q } ) + S ( x , p , \vec { Q } ) |
train/img_00000495.png | \begin{align*}k_S(x,y) := \langle S(x), S(y) \rangle_{\mathcal{H}^1}, x,y \in (\mathbb{R}^d)^I.\end{align*} |
train/img_00000496.png | \Gamma _ { V D } [ \phi ] = S [ \phi ] - \frac { 1 } { 2 } \mathrm { T r \ l n } \left[ G ^ { l i } \left( S _ { , i j } - \left\{ \begin{array} { c } { { k } } \\ { { i \ j } } \\ \end{array} \right\} S _ { , k } - T _ { i j } ^ { k } S _ { , k } \right) \right] \ , |
train/img_00000497.png | \begin{align*}J=i_\zeta(pdq-Hdt+\beta)-f=\sum_i p_i\xi_i-H\tau+\beta(\zeta)-f\end{align*} |
train/img_00000498.png | S = \frac { 1 } { \gamma ^ { 2 } } \int _ { - \infty } ^ { + \infty } d \tau \int _ { 0 } ^ { 2 \pi } d \sigma \Bigl ( \frac { 1 } { 2 } \partial _ { \alpha } \varphi \cdot \partial ^ { \alpha } \varphi - \mu ^ { 2 } \sum _ { a = 1 , 2 } \mathrm { e } ^ { \alpha ^ { a } \cdot \varphi } \Bigr ) ~ , |
train/img_00000499.png | \begin{align*}\kappa_y := \max \left( \{ \kappa(p) \mid p \Delta_y \} \right).\end{align*} |
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